Chapter 3 Widths and profiles of spectral lines Flashcards
What is the line profile
if the centre freq is v0 then the line profile is the function I(v) in the vicinity of v0
how to you get do delta lambda from the FWHM of a lien prfile
FWHM -> HWHM -> write v ito w -> w into lambda
set dlambda = |lambda1 - lambda2| and note lambda = c/v
next take d (lambda)/dv of this dont forget the sqr
What are the relative halfwidths for line profiles
|dlambda/lambda| = |dw/w| = |dv/v|
Kernal of the spectral line
Line wings
Kernel - spectral region WITHIN the halfwidth (top bump)
Wings - regions outside v1, v2 (x axis vals)
how do we desctibe the atomic electron to get natural linewidth info
model the atomic electron by the classcial model of a damped HOsc with freq w, mass m, restoring force constant k
amplitude is given by x(t)
what is gamma in the HOsc mdoel of atomic electron
the radiative enrgy loss results in the dmaping of the oscillator described by gamma (the damping constant) IRL gamma«w so for real atoms the damping is basicallly negligable
end up setting w = w0 and negling the sin term in the DE soln
Why is the frequency no longer monochromic if we model the atomic electron as a damped HOsc
The damping causes energy loss and so the amplitude x(t) of the oscillation decreases gradually
frequency is also no longer monochromatic since it will decrease also
How can the dampled oscillation x(t) be described ito combinations
A superposition of monochromatic oscillations (exp(iwt)) with slightly different freqencies w and and amplitudes A(w) -> Foureir transforms of x(t)
To get the intensity of the profile what do you need
Specifically here you want the intensity near a transition with freq w0 so I(w) -> I(w-wo)
I(w-w0) = prop A(w)A(w)*
and remeber since (w-w0)«w0, the terms with w+w0 in deno can be ignored but w-w0 cannot! (deno tens to 0) -> lorentzian profile
How do you get average power from your damped HOsc equn
If you multiply both sides bymx(dot) then you see LHS is the d/dt(potential E + Kinetic E) = RHS so whtever is on RHS (gamma damping term with mx(dot) ) IS the dW/dt
since W = total energy = PE+KE
Assuming you had eqn fro dW/dt how would you get the time averaged radiant power
radiant power = watts per unit time
= energy/time so it is dW/dt
to get the time averaged, sib in x(t) for the HOSc soln to the DE and then note the average of a sin^2 function is 1/2 that will give dWbar/dt
what useful info do you get from tiem averaged radiant power
Pbar = dWbar/dt has an exp{-gamma t}
this means the decay time T = 1/gamma
this looks same as section 2 where gamma was the eienstein A coeff for spontaneous emission
so connect the two theories to get that if you replace gamma with Ai then you get hte Correct description of the freq distribution of spont emission adn its linewidth domega
What useful info do you get from tiem averaged radiant power
Pbar = dWbar/dt has an exp{-gamma t}
this means the decay time T = 1/gamma
this looks same as section 2 where gamma was the eienstein A coeff for spontaneous emission
so connect the two theories to get that if you replace gamma with Ai then you get hte Correct description of the freq distribution of spont emission adn its linewidth domega
How does the time averaged radiant power relate to heisen berg uncetainty principle
Both describe the natural line width dw as 1/Ti
where P av suggests Ti i sthe aintein A coeff (for spont)
Relation between natural lien with and lifetime
dw = 1/Ti
so the longer the lifetime the smaller the frequency spread in the natural line width
